Gradient stability of Caffarelli-Kohn-Nirenberg inequality involving weighted p-Laplace

Abstract

The best constant and extremal functions are well known of the following Caffarelli-Kohn-Nirenberg inequality \[ \int_{\mathbb{R}^N}|\nabla u|^p\frac{\mathrm{d}x}{|x|^{\mu}}\geq \mathcal{S} \left(\int_{\mathbb{R}^N}|u|^r\frac{\mathrm{d}x}{|x|^s} \right)^{\frac{p}{r}}, \quad \mbox{for all}\quad u\in C^\infty_c(\mathbb{R}^N), \] where $1<p<p+\mu<N$, $\frac{\mu}{p}\leq \frac{s}{r}<\frac{\mu}{p}+1$, $r=\frac{p(N-s)}{N-p-\mu}$. An important task is investigating the stability of extremals for this inequality. Firstly, we give the classification to the linearized problem related to the extremals which shows the extremals are non-degenerate. Then we investigate the gradient type remainder term of previous inequality by using spectral estimate combined with a compactness argument which partially extends the work of Wei and Wu [Math. Ann., 2022] to a general $p$-Laplace case, and also the work of Figalli and Zhang [Duke Math. J., 2022] to a weighted case.